Lecture 13: Raster Graphics and Scan Conversion CS552: Computer Graphics Lecture 13: Raster Graphics and Scan Conversion

Recap Viewing pipeline Model coordinate World coordinate View coordinate Normalized coordinate Device coordinate Transformation 2D 3D Homogeneous Cliping

Objective After completing this lecture students will be able to Derive mathematical expression for drawing lines between two points Write programs to implement line drawing algorithm

2D Graphics Pipeline Simple 2D Drawing Pipeline Clipping window to Object World Coordinates Applying world window Clipping Object subset window to viewport mapping Object Screen coordinates Simple 2D Drawing Pipeline Display Rasterization

Rasterization (Scan Conversion) Convert high-level geometry description to pixel colors in the frame buffer Example: given vertex 𝑥,𝑦 coordinates determine pixel colors to draw line Two ways to create an image: Scan existing photograph Procedurally compute values (rendering) Viewport Transformation Rasterization

Rasterization A fundamental computer graphics function Determine the pixels’ colors, illuminations, textures, etc. Implemented by graphics hardware Rasterization algorithms Lines Circles Triangles Polygons

Rasterization Operations Drawing lines on the screen Manipulating pixel maps (pixmaps): copying, scaling, rotating, etc Compositing images, defining and modifying regions Drawing and filling polygons Aliasing and antialiasing methods

Line drawing algorithm Programmer specifies (x,y) values of end pixels Need algorithm to figure out which intermediate pixels are on line path Pixel (x,y) values constrained to integer values Actual computed intermediate line values may be floats Rounding may be required. E.g. computed point (10.48, 20.51) rounded to (10, 21) Rounded pixel value is off actual line path (jaggy)

Line Drawing Algorithm 0 1 2 3 4 5 6 7 8 9 10 11 12 8 7 6 5 4 3 2 1 Line: (3,2) -> (9,6) ? Which intermediate pixels to turn on?

DDA Line Drawing Algorithm (Case a: m < 1) x = x0 y = y0 Illuminate pixel (x, round(y)) (x1,y1) x = x0 + 1 y = y0 + 1 * m (x0, y0) Illuminate pixel (x, round(y)) x = x + 1 y = y + 1 * m Illuminate pixel (x, round(y)) … Until x == x1

DDA Line Drawing Algorithm (Case b: m > 1) x = x0 y = y0 (x1,y1) Illuminate pixel (round(x), y) y = y0 + 1 x = x0 + 1 * 1/m Illuminate pixel (round(x), y) y = y + 1 x = x + 1 /m Illuminate pixel (round(x), y) … (x0,y0) Until y == y1

DDA Line Drawing Algorithm Pseudocode compute m; if m < 1: { float y = y0; // initial value for(int x = x0;x <= x1; x++, y += m) setPixel(x, round(y)); } else // m > 1 float x = x0; // initial value for(int y = y0;y <= y1; y++, x += 1/m) setPixel(round(x), y); Note: setPixel(x, y) writes current color into pixel in column x and row y in frame buffer

Line Drawing Algorithm Drawbacks DDA is the simplest line drawing algorithm Not very efficient (time consuming) Round operation is expensive Optimized algorithms typically used. Integer DDA E.g. Bresenham algorithm Bresenham algorithm Incremental algorithm: current value uses previous value Integers only: avoid floating point arithmetic midpoint version of algorithm

Bresenham’s Line-Drawing Algorithm Problem: Given endpoints ( 𝑥 0 , 𝑦 0 ) and ( 𝑥 1 , 𝑦 1 ) of a line Task is to to determine best sequence of intervening pixels First make two simplifying assumptions (remove later): (0 < m < 1)

Additional Issues Endpoint order A line from 𝑃 0 to 𝑃 1 contains the same set of points as the line from 𝑃 1 to 𝑃 0 . Appearance of the line is independent of the order of the end point specification Choice of points when 𝑑=0 West or South West

Additional Issues Different slope ? Starting at the edge of a clip rectangle Different slope ? x = 𝒙 𝒎𝒊𝒏 ( 𝒙 𝒎𝒊𝒏 ,𝑹𝒐𝒖𝒏𝒅(𝒎 𝒙 𝒎 +𝑩) ( 𝑥 𝑚𝑖𝑛 ,(𝑚 𝑥 𝑚 +𝐵) y = 𝒚 𝒎𝒊𝒏

Additional Issues Different slope ? Starting at the edge of a clip rectangle Different slope ? x = 𝒙 𝒎𝒊𝒏 𝐲= 𝒚 𝒎𝒊𝒏 − 𝟏 𝟐 y = 𝒚 𝒎𝒊𝒏 𝐲= 𝒚 𝒎𝒊𝒏 −1

Additional Issues Antialiasing Line style Varying the intensity as a function of slope Antialiasing Line style

Thank you Next Lecture: Scan converting Circle